On the compactness of the Weyl operator in $\mathcal{S}_{\omega}$

We characterize, using time-frequency analysis, the continuity and compactness of the Weyl operator in global classes of ultradifferentiable functions $\mathcal{S}_\omega$, for weight functions $\omega$ in the sense of Braun, Meise and Taylor. As a consequence, we give results about the compactness of the localization operator in $\mathcal{S}_\omega$, in relation with the spaces of $\omega$-multipliers and $\omega$-convolutors of $\mathcal{S}_\omega$. Moreover, we provide several examples that complement our investigation.